Would like someone to check over my Linear algebra question

[kbrono]

Prove or disprove that each given subset of M_2x2 is a subspace of M_2x2 under the usual matrix operations.


1. The set of 2x2 matrices that have atleast one row of zeroes.

My answer: Not a subspace consider matrix A= 1 2 And matrix B= 0 0
0 0 3 0
Then A+B= 1 2
3 4

Thus not closed under addition.


2. The set of non singular 2x2 matrices

My answer: Not a subspace consider matrix A= 3 4 And matrix B= 4 6
0 1 0 -1
Then A+B= 7 10
0 0

Therefore not closed under addition

3. The set of 2x2 matrices having trace zero.

My answer: This is a subspace. Take two M_2x2 matrices A,B and a,b,c,d,e,f /in R

Then A= a b And matrix B= d e
c -a f -d

Then trace(A)=a+(-a)=0 and trace(B)=d+(-d)=0

now take A+B= a+d b+e
c+f -a-d
then trace(A+B) = a+d+(-a-d) = a-a+d-d=0 and is therefore closed under addition

Now take scalar k /in R

kA = ka kb
kc k(-a)

trace(ka)= ka+(-ak) = k(a-a) = 0. Therefore closed under scalar multiplication.

Therefore The set of 2x2 matrices having trace zero is a subset of M_2x2

 

[micromass]

Seems alright by me!